10 ms·

Teaching in general is best done by "peeling the onion". By that I mean, start with the highest level explanation first, form an understanding of that, and the

Teaching in general is best done by "peeling the onion". By that I mean, start with the highest level explanation first, form an understanding of that, and then repeat the same lesson but at a lower level of abstraction. You have to be able to walk before you can run. First, explain the concept like I'm 5. Use examples that provide real world context for why the concept is useful. Demonstrate examples of using the concept to solve a problem. Then explain the concept again, with an added layer of complexity. Go through the same examples with this layer of complexity. And so on. The more levels, the easier it will be for people to ease into learning the concept. When confronted with something that's too foreign, many people just shut off their brains, and convince themselves they're "bad at math" etc. The hardest part about learning is really breaking through the wall of initial understanding. Once you know the language, everything becomes a bit easier to learn. I honestly think math education can be hugely improved with this approach. This could even be done in a "high tech" way, by having an e-textbook with a slider. Adjusting the slider changes the level of abstraction and granularity that it teaches the concepts at. Final note. Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Or whatever, you get the point. People can't relate at all to math when all the examples use generic terms without context. At that point it just becomes rote repetition. Obviously this commentary applies moreso to earlier math education.

lixtra 6y ago
> Teaching in general is best done by "peeling the onion". My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day. The best teachers are able to adapt to that.
enriquto 6y ago
> Why use X and Y when you could carSpeed and carAcceleration? (...) People can't relate at all to math when all the examples use generic terms without context. Couldn't disagree more. The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare. Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)
- umvi 6y ago
  Variable names with several letters in them should be banished from programming languages? What do you do when you need more than 26 variables in your program?
  - sideshowb 6y ago
    Unicode bro!
- adam_arthur 6y ago
  Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material! You can apply context to any generic abstraction/formula, otherwise the math is useless, by definition. Teaching people effectively requires context. I hope I don't have to read your code with single letter variables everywhere then! There's a reason that's considered a bad practice. Math gets a pass, why?
  - enriquto 6y ago
    > Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material! I'm not sure that this kind of snark is helpful if you want to teach something. My point is precisely to make things as easy as possible, not harder! Variables are just slots. How would you rewrite the following using "meaningful" variable names? The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ? Any "meaningful" names that you could assign to the terms X, a, b, c would depend heavily in the context. You would convert this equation into an unrecognizable monstrosity. But this equation appears in many different, completely unrelated contexts! The main point of math, I'd say, is that no matter what is the context, the equation is always one and the same and it is solved in exactly the same way. Thus it pays off to have variable names as meaningless as possible.
    - 6y ago
teachingassist 6y ago
> I honestly think math education can be hugely improved with this approach. Teaching someone is more like a swiss cheese. Different students have different problems at different levels, and different holes at different levels, that they need to fill to progress more effectively. A slider (or any completely linear progression of learning) is not, in general, going to be helpful to anyone. Traditional textbooks can be of variable quality, but it's a benefit that you can flick through to any page. If a teacher can identify what page might be most useful to you, all the better. A point on a slider is unlikely to give you that. > Why use X and Y when you could carSpeed and carAcceleration? One of the purposes of math, one that makes it useful to teach to everyone universally, is that it teaches 'abstraction'. Math questions very often are about car speed and acceleration, or some other contextual example (at least in Western education systems). Using v and a to represent those makes it easier to use algebraic tools that you learn that apply generally, rather than re-learn tools for each specific situation.
- adam_arthur 6y ago
  It seems you may have misinterpreted my comment. The point is you start with context specific examples, and high level naming. Teach concepts in plain english. After which you can re-cover the same material in progressively more generic and low level forms. It's much much easier to understand the generic form of something when you actually understand contexts where it's useful, and see practical examples. I think you misunderstood my slider comment. You have a textbook, but slider adjusts the level of abstraction that it teaches you the concept at. In what way is this incompatible with flicking to a page? The TLDR; version is Explain like I'm 5, then ELI10, ELI15... and so on. There's a tendency in the hard sciences to look down on presenting concepts in a simplified form, so I understand many here will disagree with this approach. You have to struggle through the seminal 1000 page textbooks from the 70's to truly learn!
  - deleted 6y ago
    [deleted]
  - teachingassist 6y ago
    I don't think I misinterpreted your comment.
diffeomorphism 6y ago
The onions seem like the wrong metaphor here, since you are going from inside to outside. Concerning the final note: that is already the case. And then one abstracts away from that for two reasons: 1. This works just as well for boats, trucks or non-vehicles, so you use v and a. 2. Have you ever tried to write and read a formula like MassOfTheCar * VelocityOfTheCar * VelocityOfTheCar /2 =....? Notation needs to be actually usable in practice, not just pretty for single line examples.
- adam_arthur 6y ago
  > The onions seem like the wrong metaphor here, since you are going from inside to outside. I'm describing a "top down" approach, as opposed to "bottom up". Or outside in, rather than inside out. > Notation needs to be actually usable in practice, not just pretty for single line examples. Yes, math was practiced long before computers were a thing. There's a reason it's considered a bad practice to use single letter variables in code. I can assure you it's much easier to read complicated formulas in well written code than in conventional math notation! If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry. Math is no different, other than cultural biases and historical reasoning.
  - diffeomorphism 6y ago
    In math what you suggest is considered bad practice and there is also a strong reason why. Handwaving that away as "bias" or "history" is extremely rude. Both approaches have their merits and optimize for different uses. This could be an interesting discussion, but you seem uninterested in having one.
    - adam_arthur 6y ago
      I'm open to being convinced! What is the strong reason?
      - 6y ago
anonytrary 6y ago
"ELI5...N"
- adam_arthur 6y ago
  Exactly!
watwut 6y ago
> Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Definitely no beyond elementary school level and very basic concepts. If is much easier to do abstra manipulation and calculations with letters then words. And beyond very very basics, you will do a lot of that.
meristohm 6y ago
Might teaching be done Best by learning about the student(s) and figuring out what approach resonates?

lixtra 6y ago

> Teaching in general is best done by "peeling the onion". My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day. The best teachers are able to adapt to that.

enriquto 6y ago

> Why use X and Y when you could carSpeed and carAcceleration? (...) People can't relate at all to math when all the examples use generic terms without context. Couldn't disagree more. The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare. Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)

umvi 6y ago
Variable names with several letters in them should be banished from programming languages? What do you do when you need more than 26 variables in your program?
- sideshowb 6y ago
  Unicode bro!
adam_arthur 6y ago
Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material! You can apply context to any generic abstraction/formula, otherwise the math is useless, by definition. Teaching people effectively requires context. I hope I don't have to read your code with single letter variables everywhere then! There's a reason that's considered a bad practice. Math gets a pass, why?
- enriquto 6y ago
  > Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material! I'm not sure that this kind of snark is helpful if you want to teach something. My point is precisely to make things as easy as possible, not harder! Variables are just slots. How would you rewrite the following using "meaningful" variable names? The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ? Any "meaningful" names that you could assign to the terms X, a, b, c would depend heavily in the context. You would convert this equation into an unrecognizable monstrosity. But this equation appears in many different, completely unrelated contexts! The main point of math, I'd say, is that no matter what is the context, the equation is always one and the same and it is solved in exactly the same way. Thus it pays off to have variable names as meaningless as possible.
  - 6y ago

teachingassist 6y ago

> I honestly think math education can be hugely improved with this approach. Teaching someone is more like a swiss cheese. Different students have different problems at different levels, and different holes at different levels, that they need to fill to progress more effectively. A slider (or any completely linear progression of learning) is not, in general, going to be helpful to anyone. Traditional textbooks can be of variable quality, but it's a benefit that you can flick through to any page. If a teacher can identify what page might be most useful to you, all the better. A point on a slider is unlikely to give you that. > Why use X and Y when you could carSpeed and carAcceleration? One of the purposes of math, one that makes it useful to teach to everyone universally, is that it teaches 'abstraction'. Math questions very often are about car speed and acceleration, or some other contextual example (at least in Western education systems). Using v and a to represent those makes it easier to use algebraic tools that you learn that apply generally, rather than re-learn tools for each specific situation.

adam_arthur 6y ago
It seems you may have misinterpreted my comment. The point is you start with context specific examples, and high level naming. Teach concepts in plain english. After which you can re-cover the same material in progressively more generic and low level forms. It's much much easier to understand the generic form of something when you actually understand contexts where it's useful, and see practical examples. I think you misunderstood my slider comment. You have a textbook, but slider adjusts the level of abstraction that it teaches you the concept at. In what way is this incompatible with flicking to a page? The TLDR; version is Explain like I'm 5, then ELI10, ELI15... and so on. There's a tendency in the hard sciences to look down on presenting concepts in a simplified form, so I understand many here will disagree with this approach. You have to struggle through the seminal 1000 page textbooks from the 70's to truly learn!
- deleted 6y ago
  [deleted]
- teachingassist 6y ago
  I don't think I misinterpreted your comment.

diffeomorphism 6y ago

The onions seem like the wrong metaphor here, since you are going from inside to outside. Concerning the final note: that is already the case. And then one abstracts away from that for two reasons: 1. This works just as well for boats, trucks or non-vehicles, so you use v and a. 2. Have you ever tried to write and read a formula like MassOfTheCar * VelocityOfTheCar * VelocityOfTheCar /2 =....? Notation needs to be actually usable in practice, not just pretty for single line examples.

adam_arthur 6y ago
> The onions seem like the wrong metaphor here, since you are going from inside to outside. I'm describing a "top down" approach, as opposed to "bottom up". Or outside in, rather than inside out. > Notation needs to be actually usable in practice, not just pretty for single line examples. Yes, math was practiced long before computers were a thing. There's a reason it's considered a bad practice to use single letter variables in code. I can assure you it's much easier to read complicated formulas in well written code than in conventional math notation! If you were to use all one letter variable names and cram all computations into a single large expression, that would be considered by most to be poor code. There's a reason why, and it's been proven out in industry. Math is no different, other than cultural biases and historical reasoning.
- diffeomorphism 6y ago
  In math what you suggest is considered bad practice and there is also a strong reason why. Handwaving that away as "bias" or "history" is extremely rude. Both approaches have their merits and optimize for different uses. This could be an interesting discussion, but you seem uninterested in having one.
  - adam_arthur 6y ago
    I'm open to being convinced! What is the strong reason?
    - 6y ago

anonytrary 6y ago

"ELI5...N"

adam_arthur 6y ago
Exactly!

watwut 6y ago

> Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Definitely no beyond elementary school level and very basic concepts. If is much easier to do abstra manipulation and calculations with letters then words. And beyond very very basics, you will do a lot of that.

meristohm 6y ago

Might teaching be done Best by learning about the student(s) and figuring out what approach resonates?